## COMPLEX

create an ACL2 number
```Major Section:  PROGRAMMING
```

```Examples:
(complex x 3) ; x + 3i, where i is the principal square root of -1
(complex x y) ; x + yi
(complex x 0) ; same as x, for rational numbers x

```
The function `complex` takes two rational number arguments and returns an ACL2 number. This number will be of type `(complex rational)` [as defined in the Common Lisp language], except that if the second argument is zero, then `complex` returns its first argument. The function `complex-rationalp` is a recognizer for complex rational numbers, i.e. for ACL2 numbers that are not rational numbers.

The reader macro `#C` (which is the same as `#c`) provides a convenient way for typing in complex numbers. For explicit rational numbers `x` and `y`, `#C(x y)` is read to the same value as `(complex x y)`.

The functions `realpart` and `imagpart` return the real and imaginary parts (respectively) of a complex (possibly rational) number. So for example, `(realpart #C(3 4)) = 3`, `(imagpart #C(3 4)) = 4`, `(realpart 3/4) = 3/4`, and `(imagpart 3/4) = 0`.

A completion axiom that shows what `complex` returns on arguments violating its guard (which says that both arguments are rational numbers) is the following.

```(equal (complex x y)
(complex (if (rationalp x) x 0)
(if (rationalp y) y 0)))
```